Presentations of Groups with Even Length Relations

Abstract

We study the properties of groups that have presentations in which the square of each generator gives the identity and all relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a factorisation with respect to a given parabolic subgroup. Moreover, we see that cluster group presentations have even length relations and study the reduced expressions of the coset elements appearing in factorisations with respect to maximal parabolic subgroups of elements of cluster groups of mutation-Dynkin type A n .

On minimal crossing number braid diagrams and homogeneous braids

“…It is easy to see that l G,S (g[s]) = l G,S (g) ± 1 for all s ∈ S ∪ S −1 . In particular, given a geodesic word w ∈ F (S) and a letter s ∈ S ∪ S −1 , ws is geodesic if and only if s −1 ∉ R([w]) (see Lemma 3.1. in [30]). It follows that R(g) = S ∪ S −1 if and only if g is a dead end element (see [12]), that is, l G,S (gs) ≤ l G,S (g) for all s ∈ S ∪ S −1 .…”